Sums of Fibonacci numbers that are perfect powers

TL;DR

This paper investigates when sums of two Fibonacci numbers equal perfect powers, establishing that for fixed bases, solutions are extremely limited, with only a few exceptions.

Contribution

It proves the uniqueness of solutions to the sum of Fibonacci numbers equaling perfect powers for fixed bases, except for specific small cases.

Findings

At most one solution for each fixed y, except for y=2,3,4,6,10

Identifies specific exceptions where multiple solutions exist

Provides bounds and conditions for solutions to the Diophantine equation

Abstract

Let us denote by $F_n$ the $n$-th Fibonacci number. In this paper we show that for a fixed integer $y$ there exists at most one integer exponent $a>0$ such that the Diophantine equation $F_n+F_m=y^a$ has a solution $(n,m,a)$ in positive integers satisfying $n>m>0$, unless $y=2,3,4,6$ or $10$.